psychology 340 spring 2010
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Statistics for the Social Sciences. Analysis of Variance (ANOVA). Psychology 340 Spring 2010. Outline (for week). Basics of ANOVA Why Computations Post-hoc and planned comparisons Power and effect size for ANOVA Assumptions SPSS 1 factor between groups ANOVA - PowerPoint PPT PresentationTRANSCRIPT
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Analysis of Variance (ANOVA)
Statistics for the Social SciencesPsychology 340
Spring 2010
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PSY 340Statistics for the
Social SciencesOutline (for week)
• Basics of ANOVA• Why• Computations• Post-hoc and planned comparisons• Power and effect size for ANOVA• Assumptions • SPSS
– 1 factor between groups ANOVA– Post-hoc and planned comparisons
![Page 3: Psychology 340 Spring 2010](https://reader035.vdocuments.net/reader035/viewer/2022070420/56815fcb550346895dcec341/html5/thumbnails/3.jpg)
PSY 340Statistics for the
Social SciencesOutline (for week)
• Basics of ANOVA• Why• Computations• Post-hoc and planned comparisons• Power and effect size for ANOVA• Assumptions • SPSS
– 1 factor between groups ANOVA– Post-hoc and planned comparisons
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PSY 340Statistics for the
Social SciencesExample
• Effect of knowledge of prior behavior on jury decisions– Dependent variable: rate how innocent/guilty– Independent variable: 3 levels
• Criminal record• Clean record• No information (no mention of a record)
Compare the means of these three groupsClean recordJurors
Guilt Rating
Criminal record
No Information
Guilt Rating
Guilt Rating
XC
XB
XA
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PSY 340Statistics for the
Social Sciences
Statistical analysis follows design
• The 1 factor between groups ANOVA:– More than two– Independent & One
score per subject– 1 independent
variable
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PSY 340Statistics for the
Social Sciences Analysis of Variance
XB XAXC
Criminal record Clean record No information
10 5 4
7 1 6
5 3 9
10 7 3
8 4 3
XA =8.0
SSA =18.0
XB =4.0
SSB =20.0
XC =5.0
SSC =26.0
• More than two groups– Now we can’t just
compute a simple difference score since there are more than one difference
Generic test statistic
observed differencedifference expected by chance
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PSY 340Statistics for the
Social Sciences Analysis of Variance
XB XAXC
Criminal record Clean record No information
10 5 4
7 1 6
5 3 9
10 7 3
8 4 3
XA =8.0 XB =4.0 XC =5.0
SSA =18.0 SSB =20.0 SSC =26.0
– Need a measure that describes several difference scores
– Variance• Variance is essentially
an average squared difference
Observed variance
Variance from chanceF-ratio =
• More than two groups
test statistic
Tip: Many different groupings so use subscripts to keep things straight
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PSY 340Statistics for the
Social Sciences Testing Hypotheses with ANOVA
– Step 1: State your hypotheses• Hypothesis testing: a five step program
• Null hypothesis (H0)– All of the populations all have same mean
• Alternative hypotheses (HA)– Not all of the populations all have same mean– There are several alternative hypotheses– We will return to this issue later
H 0 :μA =μB =μC
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PSY 340Statistics for the
Social Sciences Testing Hypotheses with ANOVA
– Step 2: Set your decision criteria– Step 3: Collect your data – Step 4: Compute your test statistics
• Compute your estimated variances• Compute your F-ratio• Compute your degrees of freedom (there are several)
– Step 5: Make a decision about your null hypothesis
• Hypothesis testing: a five step program– Step 1: State your hypotheses
– Additional tests• Reconciling our multiple alternative hypotheses
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PSY 340Statistics for the
Social Sciences Step 4: Computing the F-ratio
• Analyzing the sources of variance– Describe the total variance in the dependent measure
• Why are these scores different?
XB XAXC
• Two sources of variability– Within groups– Between groups
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PSY 340Statistics for the
Social Sciences Step 4: Computing the F-ratio
• Within-groups estimate of the population variance – Estimating population variance from variation from
within each sample• Not affected by whether the null hypothesis is true
XB XAXC
Different people within each group
give different ratings
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PSY 340Statistics for the
Social Sciences
• Between-groups estimate of the population variance – Estimating population variance from variation between
the means of the samples• Is affected by whether the null hypothesis is true
Step 4: Computing the F-ratio
XB XAXC
There is an effectof the IV, so the
people in differentgroups give different
ratings
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PSY 340Statistics for the
Social Sciences Partitioning the variance
Total variance
Stage 1
Between groups variance
Within groups variance
Note: we will start with SS, but willget to variance
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PSY 340Statistics for the
Social Sciences Partitioning the variance
Total varianceCriminal record Clean record No information
10 5 4
7 1 6
5 3 9
10 7 3
8 4 3
GM =X∑
N=8515
=5.67
SSTotal = X −GM( )∑ 2= 10 −5.67( )2 + ...+ 3−5.67( )2 =107.33
• Basically forgetting about separate groups– Compute the
Grand Mean (GM)
– Compute squared deviations from the Grand Mean
dfTotal =N−1=15 −1=14
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PSY 340Statistics for the
Social Sciences Partitioning the variance
Total varianceCriminal record Clean record No information
10 5 4
7 1 6
5 3 9
10 7 3
8 4 3
GM =X∑
N=8515
=5.67
SSTotal = X −GM( )∑ 2= 10 −5.67( )2 + ...+ 3−5.67( )2 =107.33
• Basically forgetting about separate groups– Compute the
Grand Mean (GM)
– Compute squared deviations from the Grand Mean
dfTotal =N−1=15 −1=14
Formula alert:
SSTotal = X∑ 2−
X∑( )2
N
= 102 + 72 + ...+ 32( ) −852
15= 107.33
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PSY 340Statistics for the
Social Sciences Partitioning the variance
Total variance
Stage 1
Between groups variance
Within groups variance
SSTotal = X−GM( )∑ 2
dfTotal =N−1
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PSY 340Statistics for the
Social Sciences Partitioning the variance
Within groups varianceCriminal record Clean record No information
10 5 4
7 1 6
5 3 9
10 7 3
8 4 3
SSWithin = SSeach group∑ =SSA + SSB + SSC =64
• Basically the variability in each group1. Add up of the SS
from all of the groups
dfWithin = dfeach group∑ =4 + 4 + 4 =12
XA =8.0 XB =4.0 XC =5.0
SSA =18.0 SSB =20.0 SSC =26.0
dfA =4 dfB =4 dfC =4
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PSY 340Statistics for the
Social Sciences Partitioning the variance
Total variance
Stage 1
Between groups variance
Within groups variance
SSTotal = X−GM( )∑ 2
dfTotal =N−1
SSWithin = SSeach group∑dfWithin = dfeach group∑
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PSY 340Statistics for the
Social Sciences Partitioning the variance
Between groups varianceCriminal record Clean record No information
10 5 4
7 1 6
5 3 9
10 7 3
8 4 3
SSBetween = n X −GM( )∑ 2
• Basically how much each group differs from the Grand Mean1. Subtract the GM
from each group mean
2. Square the diffs3. Weight by number
of scores
dfbetween =#groups−1=3−1=2
XA =8.0 XB =4.0 XC =5.0
SSA =18.0 SSB =20.0 SSC =26.0
=5 8 − 5.67( )2 + 5 4 − 5.67( )2 + 5 5 − 5.67( )2
=43.3
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PSY 340Statistics for the
Social Sciences Partitioning the variance
Between groups varianceCriminal record Clean record No information
10 5 4
7 1 6
5 3 9
10 7 3
8 4 3
SSBetween = n X −GM( )∑ 2
• Basically how much each group differs from the Grand Mean1. Subtract the GM
from each group mean
2. Square the diffs3. Weight by number
of scores
dfbetween =#groups−1=3−1=2
XA =8.0 XB =4.0 XC =5.0
SSA =18.0 SSB =20.0 SSC =26.0
=5 8 − 5.67( )2 + 5 4 − 5.67( )2 + 5 5 − 5.67( )2
=43.3
Formula alert:
SSBetween =T 2
n∑ −G 2
N
=402
5+
202
5+
252
5⎛⎝⎜
⎞⎠⎟
−852
15= 43.3
T=treatment totalN=#scores in treatment
G=grand total
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PSY 340Statistics for the
Social Sciences Partitioning the variance
Total variance
Stage 1
Between groups variance
Within groups variance
SSTotal = X−GM( )∑ 2
dfTotal =N−1
SSWithin = SSeach group∑dfWithin = dfeach group∑
SSBetween = n X −GM( )∑ 2
dfbetween =#groups−1
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PSY 340Statistics for the
Social Sciences Partitioning the variance
Total variance
Stage 1
Between groups variance
Within groups variance
SSTotal = X−GM( )∑ 2
dfTotal =N−1
SSWithin = SSeach group∑dfWithin = dfeach group∑
SSBetween = n X −GM( )∑ 2
dfbetween =#groups−1
Now we return to variance. But, we call it Means Square (MS)
MSWithin =SSWithin
dfWithinMSBetween =
SSBetween
dfBetween
Recall:variance =
SSdf
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PSY 340Statistics for the
Social Sciences Partitioning the variance
Mean Squares (Variance)
SSBetween =43.3
dfbetween =2SSWithin =64dfWithin =12
MSBetween =43.32
=21.67 MSWithin =6412
=5.33
Within groups variance
Between groups variance
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PSY 340Statistics for the
Social Sciences
• The F ratio– Ratio of the between-groups to the within-groups
population variance estimate
Step 4: Computing the F-ratio
• The F distribution• The F table
Observed variance
Variance from chanceF-ratio = =
MSBetween
MSWithin
=21.675.33
= 4.07
Do we reject or failto reject the H0?
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PSY 340Statistics for the
Social Sciences Carrying out an ANOVA
• The F distribution • The F table– Need two df’s
• dfbetween (numerator)
• dfwithin (denominator)
– Values in the table correspond to critical F’s
• Reject the H0 if your computed value is greater than or equal to the critical F
– Often separate tables for 0.05 & 0.01
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PSY 340Statistics for the
Social Sciences Carrying out an ANOVA
• The F distribution • The F table– Need two df’s
• dfbetween (numerator)
• dfwithin (denominator)
– Values in the table correspond to critical F’s
• Reject the H0 if your computed value is greater than or equal to the critical F
– Often separate tables for 0.05 & 0.01
Denominator df
1 2 3 4 5 6 …
1 1624,052
2005,000
2165,404
2255,625
2305,764
2345,859
2 18.5198.50
19.099.0
19.1799.17
19.2599.25
19.3099.30
19.3399.33
3 10.1334.12
9.5530.82
9.2829.46
9.1228.71
9.0128.24
8.9427.91
4 7.7121.20
6.9518.0
6.5916.7
6.3915.98
6.2615.52
6.1615.21
5 6.6116.26
5.7913.27
5.4112.06
5.1911.39
5.0510.97
4.9510.67
6 5.9913.75
5.1410.93
4.769.78
4.539.15
4.398.75
4.288.47
…
∞
Numerator df
Table B-4, pg 731-733Lightface type are Fcrits for α = 0.05
Boldface type are Fcrits for α = 0.01
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PSY 340Statistics for the
Social Sciences Carrying out an ANOVA
• The F table– Need two df’s
• dfbetween (numerator)
• dfwithin (denominator)
– Values in the table correspond to critical F’s
• Reject the H0 if your computed value is greater than or equal to the critical F
– Often separate tables for 0.05 & 0.01
F =MSBetween
MSWithin
=21.675.33
= 4.07
Do we reject or failto reject the H0?
– From the table (assuming 0.05) with 2 and 12 degrees of freedom the critical F = 3.89.
– So we reject H0 and conclude that not all groups are the same
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PSY 340Statistics for the
Social Sciences Summary of Example ANOVA
Criminal record Clean record No information
10 5 4
7 1 6
5 3 9
10 7 3
8 4 3XA =8.0
SSA =18.0
XB =4.0
SSB =20.0
XC =5.0
SSC =26.0
GM =X∑
N=8515
=5.67
SSTotal = X −GM( )∑ 2=107.33
dfTotal =N−1=15 −1=14
SSWithin = SSeach group∑ =64
dfWithin = dfeach group∑ =4 + 4 + 4 =12
SSBetween = n X −GM( )∑ 2=43.3
dfbetween =#groups−1=3−1=2
F =MSBetween
MSWithin
=21.675.33
= 4.07
MSWithin =SSWithin
dfWithin
=5.33
MSBetween =SSBetween
dfBetween=21.67
Fcrit(2,12) = 3.89, so we reject H0
dfA =4 dfB =4 dfC =4
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PSY 340Statistics for the
Social SciencesNext time
• Basics of ANOVA• Why• Computations• Post-hoc and planned comparisons• Power and effect size for ANOVA• Assumptions • SPSS
– 1 factor between groups ANOVA– Post-hoc and planned comparisons